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Re: Derivative of implicit function

DEDONDER Philippe wrote:
> I'm a new user of Mathematica, and I would like to know if this software
> can help me in solving the following problem:
> I have a function f(t,t*,x,y)=0, and I want to compute the derivative of
> x (and also of y) with respect to t for t=t* and f(t,t*,x,y)=0. All I
> 've managed to do till now is to give particular values to t and t* and
> to Solve[f(x,y,t,t*)==0,{x,y}], which gives me particular values for x
> and y (complex as well as real values).
> Can anyone help me to solve this problem?

Without more information, the first thing that I think of is to change
your variables.  Intead of using t and t*, convert t->r Exp[j theta]
and then use r and theta, so that t=t* would be when theta=0.  Or --
use t->x+jy and then t=t* is when y=0.  The idea of taking a derivative
with respect to a complex quantity raises some questions about the best
way to evaluate it.  I think there may be some ambiguity in the way you
decide to evaluate the derivatives.

I think I would approach the derivatives by letting t*->t /
Exp[2jtheta], then taking the derivative with respect to t and setting
theta=0 at the end.

Certainly mathematica could make the substitutions for you via
transformation rules (expr/.{t->r Exp[2 I theta]}) and take the
derivatives for you via D[] and even do some solving of equations for
you via Solve[].

Good Luck.
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